Large deviation principle for the maximalpositions in the critical branching random

walks with small drifts

Hongyan Sun(China University of Geosciences, Beijing)

Beijing Normal University

8 May, 2017

Contents

1 Branching random walk

2 Critical branching random walks with small drift-s

3 Sketch of the proofs

Branching random walks

A branching random walk: an initial particle, which formsthe zeroth generation, is at the origin of R. It gives birth tooffspring particles that form the first generation. Their dis-placements from their parent are described by a point processΘ.

Branching random walks

Mn : the maximal position of the particles in the n-th generation.

Supercritical case:

Hammarsley (1974), Kingman (1975), Biggins (1976)Mnn → γ, a.s. on the non-extinction set.

Addario and Reed (2009), Hu and Shi (2009), Hu (2012,2016)established the second or the third order of Mn.

Biggins (1977) or Rouault (2000)give the large deviation principle of Mn.

Critical case: Suppose the displacements of particles from theirparents are independently and identically distributed.

Durrett (1991), Kesten (1995), Lalley and Shao (2015)(a) study the scaled limit distribution of Mn conditioned onthat the system can survive to the n-th generation.(b) give the tail distribution of M =: supn≥0Mn.

Branching random walks

Zheng (2010)studies the maximal positions of a sequence of critical branch-ing random walks with small drifts.

Subcritical case:

Neuman and Zheng (2017)(a) study the scaled limit distribution of Mn =: supk≤nMk.(b) give the tail distribution of M =: supn≥0Mn.

Our problem

The large deviation principle of the maximal positions in thecritical branching random walks with small drifts.

Critical BRWs with small drifts

For fixed n ≥ 1, we assume V (n) is the branching random walkon Z+, which starts with one particle at the origin.(A) At each time, particles produce offsprings as in a standardGalton-Waston process with mean 1 and finite variance.(B) Each particle moves one time from its parent according tothe transition probability P(α,β,n) below, where α > 0, β > 0.

P(x, x+ 1) = 12 −

βnα for x ≥ 1;

P(x, x− 1) = 12 + β

nα for x ≥ 1;P(0, 1) = 1.

(1)

V (n)(x): the position of the particle x, |x|: x’s generation.

M(n)k =: max|x|=k V

(n)(x).

Z(n)k : the number of particles at the k-th generation in V (n).

Critical BRWs with small drifts

Zheng (2010)

When α = 12 . For κ > 1, and every ε > 0,

limn→∞

P(∣∣∣ M

(n)[nκ]√

n log n− κ− 1

4β

∣∣∣ > ε| > |Z(n)[nκ] > 0

)= 0.

For any α bigger than zero, the conditioned law of large number

of M(n)[nκ] also holds with rate κ−2α

4β , for κ > 2α.

How about the large deviation principle of M(n)[nκ]?

Main result

Assume {Zm} is a critical branching process with finite variance andoffspring distribution {pj}j≥0.

Theorem 1

Assume κ > 2α. If∑j≥0 j

2(log j)pj <∞, then

(a) for any λ > κ−2α4β , we have

lim supn→∞

logP (M(n)[nκ] > λnα log n|Z(n)

[nκ] > 0)

log n

≤ max{κ− 2α− 4λβ, 2α− κ},

(b) for any κ−2α4β < λ < κ−2α

2β , we have

lim infn→∞

logP (M(n)[nκ] > λnα log n|Z(n)

[nκ] > 0)

log n

≥ κ− 2α− 4λβ.

Main result

Theorem 2

Under the assumption in Theorem 1, we have for any 0 < λ <κ−2α4β ,

lim supn→∞

logP (M(n)[nκ] ≤ λn

α log n|Z(n)[nκ] > 0)

log n≤ −(κ− 2α− 4βλ).

Corollary 3

Under the assumptions in Theorem 1, we have that for all ε > 0,

M(n)[nκ] satisfies

limn→∞

P(∣∣∣ M (n)

[nκ]

nα log n− κ− 2α

4β

∣∣∣ ≥ ε∣∣∣Z(n)[nκ] > 0

)= 0.

Idea of the proofs

Use the strategy of changing the conditional event, which isused in Zheng (2010).

Divide the branching part and the walking part.

Preparations for the proofs

Lemma 4

Assume∑

j≥0 j2(log j)pj <∞ holds. Let mn ≤ n be integers and

εn > 0 be real values such that mn/n → 1, n − mn → ∞ andεn → 0 as n→∞. Let Gn = {Zn > 0}, Hn = {Zmn ≥ nεn} andKn = {Zmn > 0}. Then(1) for all n large enough,

P (Gn∆Kn)

P (Kn)≤ δ2,

(2) for all n large enough,

P (Gn∆Hn)

P (Gn)≤ 4(δ1 + δ2),

where δ1 = Cσ2nεnmn

, δ2 = 2

1− 2σ2mn log

(1− σ2

n−mn

) .

Preparations for the proofs

Let {S(n)k } denote a random walk with transition probability

P(α,β,n) given by (1).

Proposition 5

For any fixed b > 0, and any nonnegative sequences {ln}, {kn}with lim supn→∞ ln/n

α <∞ and lim infn→∞ kn/(n2α log2 n) > 0,

the random walks {S(n)k } satisfy

limn→∞

P (S(n)kn≥ bnα|S(n)

0 = ln) = exp(−4βb),

and

limn→∞

P (S(n)kn≥ bnα log n|S(n)

0 = ln)

n−4βb= 1.

Proof of Theorem 1 (a)

For all n large enough

|P (M(n)[nκ] ≥ λn

α log n|Z(n)[nκ] > 0)− P (M

(n)[nκ] ≥ λn

α log n|Z(n)[nκ]−[n2α] > 0)|

≤ 4n2α−κ (by Lemma 4). (2)

We divide the probability space into two event. And can get that

P (M(n)[nκ] ≥ λn

α log n,M(n)[nκ]−[n2α] ≥ (λ− ε)nα log n|Z(n)

[nκ]−[n2α] > 0)

≤ E(Z(n)[nκ]−[n2α]|Z

(n)[nκ]−[n2α] > 0)× P (S

(α,β,n)[nκ]−[n2α] > (λ− ε)nα log n)× ρ[n2α]

≤ σ2(nκ − n2α)× 2n−4β(λ−ε) × 4

σ2(n2α)≤ 8nκ−4βλ−2α+4βε.

Proof of Theorem 1 (a)

Let M′(n)nκ denote the rightmost position of the particles in [nκ]-th gen-

eration whose ancestors in ([nκ]− [n2α])-th generation is in the left of(λ− ε)nα log n.

Since the branching random walk is critical, we get for all n largeenough

P (M′(n)[nκ] ≥ λn

α log n|Z(n)[nκ]−[n2α] > 0)

≤ E(Z(n)[nκ]−[n2α]|Z

(n)[nκ]−[n2α] > 0)

·P (S[nκ] > λnα log n|S[nκ]−[n2α] < (λ− ε)nα log n)

≤ σ2(nκ − n2α) · C1n−bε2 logn ≤ C1σ

2n−(bε2(logn)2−κ),

where C1 and b are constants irrelevant to n.

Combining these two situations, we have

P (M(n)nκ ≥ λnα log n|Z(n)

nκ−n2α > 0) ≤ 8nκ−4βλ−2α+4βε+Cσ2n−(bε2(logn)2−κ).

Proof of Theorem 1 (a)

Then by (2), we get that for all n large enough

P (M(n)nκ ≥ λnα log n|Z(n)

nκ > 0) ≤ 8nκ−4βλ−2α+4βε + Cσ2n−(bε2(logn)2−κ)

+2n2α−κ.

Therefore, we have that for any ε > 0,

lim supn→∞

logP (Mnκ ≥ λnα log n|Z(n)[nκ] > 0)

log n≤ max{κ−2α−4βλ+4βε, 2α−κ}.

Reference

[1] Addario-Berry, L., Reed, B. (2009). Minima in branching randomwalks. Ann. Prob. 37 1044-1079.

[2] Biggins, J. D. (1976). The first and last birth problems for amultitype age-dependent branching process. Adv. Appl. Probab.8 446-459.

[3] Durrett, R., Kesten, H., Waymire,E. (1991). On weighted heightsof random trees. J.Theoret.Prob. 4 223-237.

[4] Neuman, E. Zheng X.H. (2017). On the maximal displacement ofsubcritical branching random walks, Probab. Related Fields. 1671137-1164.

[5] Hu, Y. (2012). The almost sure limits of the minimal position andthe additive martingale in a branching random walk. J.Theor.Probab.28 467-487.

[6] Hu, Y. Y. (2016). How big the minimum of a branching randomwalk. Ann. Inst. H. Poincare Probab. Statist. 52 233-260.

[7] Hu, Y., Shi, Z. (2009). Minimal position and critical martingaleconvergence in branching random walks, and directed polymerson disordered trees. Ann.Probab. 37 742-789.

Reference

[8] Kesten, H. (1995). Branching random walk with a critical branch-ing part. J. Theoret.Prob. 8 921-962.

[9] Lalley, S. P., Shao, Y. (2015). On the maximal displacement ofcritical branching random walk. Probability Theory and RelatedFields 162 71-96.

[10] Zheng, X. H. (2010). Critical branching random walks with smalldrift. Stoch.Proc.Appl. 120 1821-1836.

Acknowledgement

Thanks for your attention!